Tablas estadísticas_Walpole

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726 Apéndice A Tablas y demostraciones estadísticas
Tabla A.1 Sumas de probabilidad binomial b(x; n, p)
r
∑
x= 0
p
 n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
 1 0 0.9000 0.8000 0.7500 0.7000 0.6000 0.5000 0.4000 0.3000 0.2000 0.1000
 1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
 2 0 0.8100 0.6400 0.5625 0.4900 0.3600 0.2500 0.1600 0.0900 0.0400 0.0100
 1 0.9900 0.9600 0.9375 0.9100 0.8400 0.7500 0.6400 0.5100 0.3600 0.1900
 2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
 3 0 0.7290 0.5120 0.4219 0.3430 0.2160 0.1250 0.0640 0.0270 0.0080 0.0010
 1 0.9720 0.8960 0.8438 0.7840 0.6480 0.5000 0.3520 0.2160 0.1040 0.0280
 2 0.9990 0.9920 0.9844 0.9730 0.9360 0.8750 0.7840 0.6570 0.4880 0.2710
 3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
 4 0 0.6561 0.4096 0.3164 0.2401 0.1296 0.0625 0.0256 0.0081 0.0016 0.0001
 1 0.9477 0.8192 0.7383 0.6517 0.4752 0.3125 0.1792 0.0837 0.0272 0.0037
 2 0.9963 0.9728 0.9492 0.9163 0.8208 0.6875 0.5248 0.3483 0.1808 0.0523
 3 0.9999 0.9984 0.9961 0.9919 0.9744 0.9375 0.8704 0.7599 0.5904 0.3439
 4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
 5 0 0.5905 0.3277 0.2373 0.1681 0.0778 0.0313 0.0102 0.0024 0.0003 0.0000
 1 0.9185 0.7373 0.6328 0.5282 0.3370 0.1875 0.0870 0.0308 0.0067 0.0005
 2 0.9914 0.9421 0.8965 0.8369 0.6826 0.5000 0.3174 0.1631 0.0579 0.0086
 3 0.9995 0.9933 0.9844 0.9692 0.9130 0.8125 0.6630 0.4718 0.2627 0.0815
 4 1.0000 0.9997 0.9990 0.9976 0.9898 0.9688 0.9222 0.8319 0.6723 0.4095
 5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
 6 0 0.5314 0.2621 0.1780 0.1176 0.0467 0.0156 0.0041 0.0007 0.0001 0.0000
 1 0.8857 0.6554 0.5339 0.4202 0.2333 0.1094 0.0410 0.0109 0.0016 0.0001
 2 0.9842 0.9011 0.8306 0.7443 0.5443 0.3438 0.1792 0.0705 0.0170 0.0013
 3 0.9987 0.9830 0.9624 0.9295 0.8208 0.6563 0.4557 0.2557 0.0989 0.0159
 4 0.9999 0.9984 0.9954 0.9891 0.9590 0.8906 0.7667 0.5798 0.3446 0.1143
 5 1.0000 0.9999 0.9998 0.9993 0.9959 0.9844 0.9533 0.8824 0.7379 0.4686
 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
 7 0 0.4783 0.2097 0.1335 0.0824 0.0280 0.0078 0.0016 0.0002 0.0000
 1 0.8503 0.5767 0.4449 0.3294 0.1586 0.0625 0.0188 0.0038 0.0004 0.0000
 2 0.9743 0.8520 0.7564 0.6471 0.4199 0.2266 0.0963 0.0288 0.0047 0.0002
 3 0.9973 0.9667 0.9294 0.8740 0.7102 0.5000 0.2898 0.1260 0.0333 0.0027
 4 0.9998 0.9953 0.9871 0.9712 0.9037 0.7734 0.5801 0.3529 0.1480 0.0257
 5 1.0000 0.9996 0.9987 0.9962 0.9812 0.9375 0.8414 0.6706 0.4233 0.1497
 6 1.0000 0.9999 0.9998 0.9984 0.9922 0.9720 0.9176 0.7903 0.5217
 7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Tabla A.1 Sumas de probabilidad binomial 727
Tabla A.1 (continuación) Sumas de probabilidad binomial b(x; n, p)
r
∑
x= 0
p
 n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
 8 0 0.4305 0.1678 0.1001 0.0576 0.0168 0.0039 0.0007 0.0001 0.0000
 1 0.8131 0.5033 0.3671 0.2553 0.1064 0.0352 0.0085 0.0013 0.0001
 2 0.9619 0.7969 0.6785 0.5518 0.3154 0.1445 0.0498 0.0113 0.0012 0.0000
 3 0.9950 0.9437 0.8862 0.8059 0.5941 0.3633 0.1737 0.0580 0.0104 0.0004
 4 0.9996 0.9896 0.9727 0.9420 0.8263 0.6367 0.4059 0.1941 0.0563 0.0050
 5 1.0000 0.9988 0.9958 0.9887 0.9502 0.8555 0.6846 0.4482 0.2031 0.0381
 6 0.9999 0.9996 0.9987 0.9915 0.9648 0.8936 0.7447 0.4967 0.1869
 7 1.0000 1.0000 0.9999 0.9993 0.9961 0.9832 0.9424 0.8322 0.5695
 8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
 9 0 0.3874 0.1342 0.0751 0.0404 0.0101 0.0020 0.0003 0.0000
 1 0.7748 0.4362 0.3003 0.1960 0.0705 0.0195 0.0038 0.0004 0.0000
 2 0.9470 0.7382 0.6007 0.4628 0.2318 0.0898 0.0250 0.0043 0.0003 0.0000
 3 0.9917 0.9144 0.8343 0.7297 0.4826 0.2539 0.0994 0.0253 0.0031 0.0001
 4 0.9991 0.9804 0.9511 0.9012 0.7334 0.5000 0.2666 0.0988 0.0196 0.0009
 5 0.9999 0.9969 0.9900 0.9747 0.9006 0.7461 0.5174 0.2703 0.0856 0.0083
 6 1.0000 0.9997 0.9987 0.9957 0.9750 0.9102 0.7682 0.5372 0.2618 0.0530
 7 1.0000 0.9999 0.9996 0.9962 0.9805 0.9295 0.8040 0.5638 0.2252
 8 1.0000 1.0000 0.9997 0.9980 0.9899 0.9596 0.8658 0.6126
 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
 10 0 0.3487 0.1074 0.0563 0.0282 0.0060 0.0010 0.0001 0.0000
 1 0.7361 0.3758 0.2440 0.1493 0.0464 0.0107 0.0017 0.0001 0.0000
 2 0.9298 0.6778 0.5256 0.3828 0.1673 0.0547 0.0123 0.0016 0.0001
 3 0.9872 0.8791 0.7759 0.6496 0.3823 0.1719 0.0548 0.0106 0.0009 0.0000
 4 0.9984 0.9672 0.9219 0.8497 0.6331 0.3770 0.1662 0.0473 0.0064 0.0001
 5 0.9999 0.9936 0.9803 0.9527 0.8338 0.6230 0.3669 0.1503 0.0328 0.0016
 6 1.0000 0.9991 0.9965 0.9894 0.9452 0.8281 0.6177 0.3504 0.1209 0.0128
 7 0.9999 0.9996 0.9984 0.9877 0.9453 0.8327 0.6172 0.3222 0.0702
 8 1.0000 1.0000 0.9999 0.9983 0.9893 0.9536 0.8507 0.6242 0.2639
 9 1.0000 0.9999 0.9990 0.9940 0.9718 0.8926 0.6513
 10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
 11 0 0.3138 0.0859 0.0422 0.0198 0.0036 0.0005 0.0000
 1 0.6974 0.3221 0.1971 0.1130 0.0302 0.0059 0.0007 0.0000
 2 0.9104 0.6174 0.4552 0.3127 0.1189 0.0327 0.0059 0.0006 0.0000
 3 0.9815 0.8389 0.7133 0.5696 0.2963 0.1133 0.0293 0.0043 0.0002
 4 0.9972 0.9496 0.8854 0.7897 0.5328 0.2744 0.0994 0.0216 0.0020 0.0000
 5 0.9997 0.9883 0.9657 0.9218 0.7535 0.5000 0.2465 0.0782 0.0117 0.0003
 6 1.0000 0.9980 0.9924 0.9784 0.9006 0.7256 0.4672 0.2103 0.0504 0.0028
 7 0.9998 0.9988 0.9957 0.9707 0.8867 0.7037 0.4304 0.1611 0.0185
 8 1.0000 0.9999 0.9994 0.9941 0.9673 0.8811 0.6873 0.3826 0.0896
 9 1.0000 1.0000 0.9993 0.9941 0.9698 0.8870 0.6779 0.3026
 10 1.0000 0.9995 0.9964 0.9802 0.9141 0.6862
 11 1.0000 1.0000 1.0000 1.0000 1.0000
728 Apéndice A Tablas y demostraciones estadísticas
Tabla A.1 (continuación) Sumas de probabilidad binomial b(x; n, p)
r
∑
x= 0
p
 n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
 12 0 0.2824 0.0687 0.0317 0.0138 0.0022 0.0002 0.0000
 1 0.6590 0.2749 0.1584 0.0850 0.0196 0.0032 0.0003 0.0000
 2 0.8891 0.5583 0.3907 0.2528 0.0834 0.0193 0.0028 0.0002 0.0000
 3 0.9744 0.7946 0.6488 0.4925 0.2253 0.0730 0.0153 0.0017 0.0001
 4 0.9957 0.9274 0.8424 0.7237 0.4382 0.1938 0.0573 0.0095 0.0006 0.0000
 5 0.9995 0.9806 0.9456 0.8822 0.6652 0.3872 0.1582 0.0386 0.0039 0.0001
 6 0.9999 0.9961 0.9857 0.9614 0.8418 0.6128 0.3348 0.1178 0.0194 0.0005
 7 1.0000 0.9994 0.9972 0.9905 0.9427 0.8062 0.5618 0.2763 0.0726 0.0043
 8 0.9999 0.9996 0.9983 0.9847 0.9270 0.7747 0.5075 0.2054 0.0256
 9 1.0000 1.0000 0.9998 0.9972 0.9807 0.9166 0.7472 0.4417 0.1109
 10 1.0000 0.9997 0.9968 0.9804 0.9150 0.7251 0.3410
 11 1.0000 0.9998 0.9978 0.9862 0.9313 0.7176
 12 1.0000 1.0000 1.0000 1.0000 1.0000
 13 0 0.2542 0.0550 0.0238 0.0097 0.0013 0.0001 0.0000
 1 0.6213 0.2336 0.1267 0.0637 0.0126 0.0017 0.0001 0.0000
 2 0.8661 0.5017 0.3326 0.2025 0.0579 0.0112 0.0013 0.0001
 3 0.9658 0.7473 0.5843 0.4206 0.1686 0.0461 0.0078 0.0007 0.0000
 4 0.9935 0.9009 0.7940 0.6543 0.3530 0.1334 0.0321 0.0040 0.0002
 5 0.9991 0.9700 0.9198 0.8346 0.5744 0.2905 0.0977 0.0182 0.0012 0.0000
 6 0.9999 0.9930 0.9757 0.9376 0.7712 0.5000 0.2288 0.0624 0.0070 0.0001
 7 1.0000 0.9988 0.9944 0.9818 0.9023 0.7095 0.4256 0.1654 0.0300 0.0009
 8 0.9998 0.9990 0.9960 0.9679 0.8666 0.6470 0.3457 0.0991 0.0065
 9 1.0000 0.9999 0.9993 0.9922 0.9539 0.8314 0.5794 0.2527 0.0342
 10 1.0000 0.9999 0.9987 0.9888 0.9421 0.7975 0.4983 0.1339
 11 1.0000 0.9999 0.9983 0.9874 0.9363 0.7664 0.3787
 12 1.0000 0.9999 0.9987 0.9903 0.9450 0.7458
 13 1.0000 1.0000 1.0000 1.0000 1.0000
 14 0 0.2288 0.0440 0.0178 0.0068 0.0008 0.0001 0.0000
 1 0.5846 0.1979 0.1010 0.0475 0.0081 0.0009 0.0001
 2 0.8416 0.4481 0.2811 0.1608 0.0398 0.0065 0.0006 0.0000
 3 0.9559 0.6982 0.5213 0.3552 0.1243 0.0287 0.0039 0.0002
 4 0.9908 0.8702 0.7415 0.5842 0.2793 0.0898 0.0175 0.00170.0000
 5 0.9985 0.9561 0.8883 0.7805 0.4859 0.2120 0.0583 0.0083 0.0004
 6 0.9998 0.9884 0.9617 0.9067 0.6925 0.3953 0.1501 0.0315 0.0024 0.0000
 7 1.0000 0.9976 0.9897 0.9685 0.8499 0.6047 0.3075 0.0933 0.0116 0.0002
 8 0.9996 0.9978 0.9917 0.9417 0.7880 0.5141 0.2195 0.0439 0.0015
 9 1.0000 0.9997 0.9983 0.9825 0.9102 0.7207 0.4158 0.1298 0.0092
 10 1.0000 0.9998 0.9961 0.9713 0.8757 0.6448 0.3018 0.0441
 11 1.0000 0.9994 0.9935 0.9602 0.8392 0.5519 0.1584
 12 0.9999 0.9991 0.9919 0.9525 0.8021 0.4154
 13 1.0000 0.9999 0.9992 0.9932 0.9560 0.7712
 14 1.0000 1.0000 1.0000 1.0000 1.0000
Tabla A.1 Sumas de probabilidad binomial 729
Tabla A.1 (continuación) Sumas de probabilidad binomial b(x; n, p)
r
∑
x= 0
p
 n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
 15 0 0.2059 0.0352 0.0134 0.0047 0.0005 0.0000
 1 0.5490 0.1671 0.0802 0.0353 0.0052 0.0005 0.0000
 2 0.8159 0.3980 0.2361 0.1268 0.0271 0.0037 0.0003 0.0000
 3 0.9444 0.6482 0.4613 0.2969 0.0905 0.0176 0.0019 0.0001
 4 0.9873 0.8358 0.6865 0.5155 0.2173 0.0592 0.0093 0.0007 0.0000
 5 0.9978 0.9389 0.8516 0.7216 0.4032 0.1509 0.0338 0.0037 0.0001
 6 0.9997 0.9819 0.9434 0.8689 0.6098 0.3036 0.0950 0.0152 0.0008
 7 1.0000 0.9958 0.9827 0.9500 0.7869 0.5000 0.2131 0.0500 0.0042 0.0000
 8 0.9992 0.9958 0.9848 0.9050 0.6964 0.3902 0.1311 0.0181 0.0003
 9 0.9999 0.9992 0.9963 0.9662 0.8491 0.5968 0.2784 0.0611 0.0022
 10 1.0000 0.9999 0.9993 0.9907 0.9408 0.7827 0.4845 0.1642 0.0127
 11 1.0000 0.9999 0.9981 0.9824 0.9095 0.7031 0.3518 0.0556
 12 1.0000 0.9997 0.9963 0.9729 0.8732 0.6020 0.1841
 13 1.0000 0.9995 0.9948 0.9647 0.8329 0.4510
 14 1.0000 0.9995 0.9953 0.9648 0.7941
 15 1.0000 1.0000 1.0000 1.0000
 16 0 0.1853 0.0281 0.0100 0.0033 0.0003 0.0000
 1 0.5147 0.1407 0.0635 0.0261 0.0033 0.0003 0.0000
 2 0.7892 0.3518 0.1971 0.0994 0.0183 0.0021 0.0001
 3 0.9316 0.5981 0.4050 0.2459 0.0651 0.0106 0.0009 0.0000
 4 0.9830 0.7982 0.6302 0.4499 0.1666 0.0384 0.0049 0.0003
 5 0.9967 0.9183 0.8103 0.6598 0.3288 0.1051 0.0191 0.0016 0.0000
 6 0.9995 0.9733 0.9204 0.8247 0.5272 0.2272 0.0583 0.0071 0.0002
 7 0.9999 0.9930 0.9729 0.9256 0.7161 0.4018 0.1423 0.0257 0.0015 0.0000
 8 1.0000 0.9985 0.9925 0.9743 0.8577 0.5982 0.2839 0.0744 0.0070 0.0001
 9 0.9998 0.9984 0.9929 0.9417 0.7728 0.4728 0.1753 0.0267 0.0005
 10 1.0000 0.9997 0.9984 0.9809 0.8949 0.6712 0.3402 0.0817 0.0033
 11 1.0000 0.9997 0.9951 0.9616 0.8334 0.5501 0.2018 0.0170
 12 1.0000 0.9991 0.9894 0.9349 0.7541 0.4019 0.0684
 13 0.9999 0.9979 0.9817 0.9006 0.6482 0.2108
 14 1.0000 0.9997 0.9967 0.9739 0.8593 0.4853
 15 1.0000 0.9997 0.9967 0.9719 0.8147
 16 1.0000 1.0000 1.0000 1.0000
730 Apéndice A Tablas y demostraciones estadísticas
Tabla A.1 (continuación) Sumas de probabilidad binomial b(x; n, p)
r
∑
x= 0
p
 n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
 17 0 0.1668 0.0225 0.0075 0.0023 0.0002 0.0000
 1 0.4818 0.1182 0.0501 0.0193 0.0021 0.0001 0.0000
 2 0.7618 0.3096 0.1637 0.0774 0.0123 0.0012 0.0001
 3 0.9174 0.5489 0.3530 0.2019 0.0464 0.0064 0.0005 0.0000
 4 0.9779 0.7582 0.5739 0.3887 0.1260 0.0245 0.0025 0.0001
 5 0.9953 0.8943 0.7653 0.5968 0.2639 0.0717 0.0106 0.0007 0.0000
 6 0.9992 0.9623 0.8929 0.7752 0.4478 0.1662 0.0348 0.0032 0.0001
 7 0.9999 0.9891 0.9598 0.8954 0.6405 0.3145 0.0919 0.0127 0.0005
 8 1.0000 0.9974 0.9876 0.9597 0.8011 0.5000 0.1989 0.0403 0.0026 0.0000
 9 0.9995 0.9969 0.9873 0.9081 0.6855 0.3595 0.1046 0.0109 0.0001
 10 0.9999 0.9994 0.9968 0.9652 0.8338 0.5522 0.2248 0.0377 0.0008
 11 1.0000 0.9999 0.9993 0.9894 0.9283 0.7361 0.4032 0.1057 0.0047
 12 1.0000 0.9999 0.9975 0.9755 0.8740 0.6113 0.2418 0.0221
 13 1.0000 0.9995 0.9936 0.9536 0.7981 0.4511 0.0826
 14 0.9999 0.9988 0.9877 0.9226 0.6904 0.2382
 15 1.0000 0.9999 0.9979 0.9807 0.8818 0.5182
 16 1.0000 0.9998 0.9977 0.9775 0.8332
 17 1.0000 1.0000 1.0000 1.0000
18 0 0.1501 0.0180 0.0056 0.0016 0.0001 0.0000
 1 0.4503 0.0991 0.0395 0.0142 0.0013 0.0001
 2 0.7338 0.2713 0.1353 0.0600 0.0082 0.0007 0.0000
 3 0.9018 0.5010 0.3057 0.1646 0.0328 0.0038 0.0002
 4 0.9718 0.7164 0.5187 0.3327 0.0942 0.0154 0.0013 0.0000
 5 0.9936 0.8671 0.7175 0.5344 0.2088 0.0481 0.0058 0.0003
 6 0.9988 0.9487 0.8610 0.7217 0.3743 0.1189 0.0203 0.0014 0.0000
 7 0.9998 0.9837 0.9431 0.8593 0.5634 0.2403 0.0576 0.0061 0.0002
 8 1.0000 0.9957 0.9807 0.9404 0.7368 0.4073 0.1347 0.0210 0.0009
 9 0.9991 0.9946 0.9790 0.8653 0.5927 0.2632 0.0596 0.0043 0.0000
 10 0.9998 0.9988 0.9939 0.9424 0.7597 0.4366 0.1407 0.0163 0.0002
 11 1.0000 0.9998 0.9986 0.9797 0.8811 0.6257 0.2783 0.0513 0.0012
 12 1.0000 0.9997 0.9942 0.9519 0.7912 0.4656 0.1329 0.0064
 13 1.0000 0.9987 0.9846 0.9058 0.6673 0.2836 0.0282
 14 0.9998 0.9962 0.9672 0.8354 0.4990 0.0982
 15 1.0000 0.9993 0.9918 0.9400 0.7287 0.2662
 16 0.9999 0.9987 0.9858 0.9009 0.5497
 17 1.0000 0.9999 0.9984 0.9820 0.8499
 18 1.0000 1.0000 1.0000 1.0000
Tabla A.1 Sumas de probabilidad binomial 731
Tabla A.1 (continuación) Sumas de probabilidad binomial b(x; n, p)
r
∑
x= 0
p
 n r 0.10 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90
 19 0 0.1351 0.0144 0.0042 0.0011 0.0001
 1 0.4203 0.0829 0.0310 0.0104 0.0008 0.0000
 2 0.7054 0.2369 0.1113 0.0462 0.0055 0.0004 0.0000
 3 0.8850 0.4551 0.2631 0.1332 0.0230 0.0022 0.0001
 4 0.9648 0.6733 0.4654 0.2822 0.0696 0.0096 0.0006 0.0000
 5 0.9914 0.8369 0.6678 0.4739 0.1629 0.0318 0.0031 0.0001
 6 0.9983 0.9324 0.8251 0.6655 0.3081 0.0835 0.0116 0.0006
 7 0.9997 0.9767 0.9225 0.8180 0.4878 0.1796 0.0352 0.0028 0.0000
 8 1.0000 0.9933 0.9713 0.9161 0.6675 0.3238 0.0885 0.0105 0.0003
 9 0.9984 0.9911 0.9674 0.8139 0.5000 0.1861 0.0326 0.0016
 10 0.9997 0.9977 0.9895 0.9115 0.6762 0.3325 0.0839 0.0067 0.0000
 11 1.0000 0.9995 0.9972 0.9648 0.8204 0.5122 0.1820 0.0233 0.0003
 12 0.9999 0.9994 0.9884 0.9165 0.6919 0.3345 0.0676 0.0017
 13 1.0000 0.9999 0.9969 0.9682 0.8371 0.5261 0.1631 0.0086
 14 1.0000 0.9994 0.9904 0.9304 0.7178 0.3267 0.0352
 15 0.9999 0.9978 0.9770 0.8668 0.5449 0.1150
 16 1.0000 0.9996 0.9945 0.9538 0.7631 0.2946
 17 1.0000 0.9992 0.9896 0.9171 0.5797
 18 0.9999 0.9989 0.9856 0.8649
 19 1.0000 1.0000 1.0000 1.0000
 20 0 0.1216 0.0115 0.0032 0.0008 0.0000
 1 0.3917 0.0692 0.0243 0.0076 0.0005 0.0000
 2 0.6769 0.2061 0.0913 0.0355 0.0036 0.0002
 3 0.8670 0.4114 0.2252 0.1071 0.0160 0.0013 0.0000
 4 0.9568 0.6296 0.4148 0.2375 0.0510 0.0059 0.0003
 5 0.9887 0.8042 0.6172 0.4164 0.1256 0.0207 0.0016 0.0000
 6 0.9976 0.9133 0.7858 0.6080 0.2500 0.0577 0.0065 0.0003
 7 0.9996 0.9679 0.8982 0.7723 0.4159 0.1316 0.0210 0.0013 0.0000
 8 0.9999 0.9900 0.9591 0.8867 0.5956 0.2517 0.0565 0.0051 0.0001
 9 1.0000 0.9974 0.9861 0.9520 0.7553 0.4119 0.1275 0.0171 0.0006
 10 0.9994 0.9961 0.9829 0.8725 0.5881 0.2447 0.0480 0.0026 0.0000
 11 0.9999 0.9991 0.9949 0.9435 0.7483 0.4044 0.1133 0.0100 0.0001
 12 1.0000 0.9998 0.9987 0.9790 0.8684 0.5841 0.2277 0.0321 0.0004
 13 1.0000 0.9997 0.9935 0.9423 0.7500 0.3920 0.0867 0.0024
 14 1.0000 0.9984 0.9793 0.8744 0.5836 0.1958 0.0113
 15 0.9997 0.9941 0.9490 0.7625 0.3704 0.0432
 16 1.0000 0.9987 0.9840 0.8929 0.5886 0.1330
 17 0.9998 0.9964 0.9645 0.7939 0.3231
 18 1.0000 0.9995 0.9924 0.9308 0.6083
 19 1.0000 0.9992 0.9885 0.8784
 20 1.0000 1.0000 1.0000
732 Apéndice A Tablas y demostraciones estadísticas
Tabla A.2 Sumas de probabilidad de Poisson p(x; µ)
x = 0
r
∑
µ
 r 0.1 0.2 0.30 0.4 0.5 0.6 0.7 0.8 0.9
 0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066
 1 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.7725
 2 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.9371
 3 1.0000 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.9865
 4 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0.9977
 5 1.00001.0000 1.0000 0.9999 0.9998 0.9997
 6 1.0000 1.0000 1.0000
µ
 r 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
 0 0.3679 0.2231 0.1353 0.0821 0.0498 0.0302 0.0183 0.0111 0.0067
 1 0.7358 0.5578 0.4060 0.2873 0.1991 0.1359 0.0916 0.0611 0.0404
 2 0.9197 0.8088 0.6767 0.5438 0.4232 0.3208 0.2381 0.1736 0.1247
 3 0.9810 0.9344 0.8571 0.7576 0.6472 0.5366 0.4335 0.3423 0.2650
 4 0.9963 0.9814 0.9473 0.8912 0.8153 0.7254 0.6288 0.5321 0.4405
 5 0.9994 0.9955 0.9834 0.9580 0.9161 0.8576 0.7851 0.7029 0.6160
 6 0.9999 0.9991 0.9955 0.9858 0.9665 0.9347 0.8893 0.8311 0.7622
 7 1.0000 0.9998 0.9989 0.9958 0.9881 0.9733 0.9489 0.9134 0.8666
 8 1.0000 0.9998 0.9989 0.9962 0.9901 0.9786 0.9597 0.9319
 9 1.0000 0.9997 0.9989 0.9967 0.9919 0.9829 0.9682
 10 0.9999 0.9997 0.9990 0.9972 0.9933 0.9863
 11 1.0000 0.9999 0.9997 0.9991 0.9976 0.9945
 12 1.0000 0.9999 0.9997 0.9992 0.9980
 13 1.0000 0.9999 0.9997 0.9993
 14 1.0000 0.9999 0.9998
 15 1.0000 0.9999
 16 1.0000
Tabla A.2 Sumas de probabilidad de Poisson 733
Tabla A.2 (continuación) Sumas de probabilidad de Poisson p(x; µ)
x = 0
r
∑
µ
 r 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5
 0 0.0041 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0.0001
 1 0.0266 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0.0008
 2 0.0884 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0.0042
 3 0.2017 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0.0149
 4 0.3575 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0.0403
 5 0.5289 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0.0885
 6 0.6860 0.6063 0.5265 0.4497 0.3782 0.3134 0.2562 0.2068 0.1649
 7 0.8095 0.7440 0.6728 0.5987 0.5246 0.4530 0.3856 0.3239 0.2687
 8 0.8944 0.8472 0.7916 0.7291 0.6620 0.5925 0.5231 0.4557 0.3918
 9 0.9462 0.9161 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218
 10 0.9747 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0.6453
 11 0.9890 0.9799 0.9661 0.9467 0.9208 0.8881 0.8487 0.8030 0.7520
 12 0.9955 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0.8364
 13 0.9983 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0.8981
 14 0.9994 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0.9400
 15 0.9998 0.9995 0.9988 0.9976 0.9954 0.9918 0.9862 0.9780 0.9665
 16 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.9823
 17 1.0000 0.9999 0.9998 0.9996 0.9992 0.9984 0.9970 0.9947 0.9911
 18 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.9957
 19 1.0000 1.0000 0.9999 0.9997 0.9995 0.9989 0.9980
 20 0.9999 0.9998 0.9996 0.9991
 21 1.0000 0.9999 0.9998 0.9996
 22 1.0000 0.9999 0.9999
 23 1.0000 0.9999
 24 1.0000
734 Apéndice A Tablas y demostraciones estadísticas
Tabla A.2 (continuación) Sumas de probabilidad de Poisson p(x; µ)
x = 0
r
∑
µ
 r 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0
 0 0.0000 0.0000 0.0000
 1 0.0005 0.0002 0.0001 0.0000 0.0000
 2 0.0028 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000
 3 0.0103 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000
 4 0.0293 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001
 5 0.0671 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003
 6 0.1301 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010
 7 0.2202 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029
 8 0.3328 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.0071
 9 0.4579 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154
 10 0.5830 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304
 11 0.6968 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549
 12 0.7916 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917
 13 0.8645 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.1426
 14 0.9165 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.2081
 15 0.9513 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867
 16 0.9730 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751
 17 0.9857 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686
 18 0.9928 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622
 19 0.9965 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509
 20 0.9984 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307
 21 0.9993 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.7991
 22 0.9997 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551
 23 0.9999 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989
 24 1.0000 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317
 25 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554
 26 1.0000 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.9718
 27 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.9827
 28 1.0000 0.9999 0.9997 0.9991 0.9978 0.9950 0.9897
 29 1.0000 0.9999 0.9996 0.9989 0.9973 0.9941
 30 0.9999 0.9998 0.9994 0.9986 0.9967
 31 1.0000 0.9999 0.9997 0.9993 0.9982
 32 1.0000 0.9999 0.9996 0.9990
 33 0.9999 0.9998 0.9995
 34 1.0000 0.9999 0.9998
 35 1.0000 0.9999
 36 0.9999
 37 1.0000
Tabla A.3 Áreas bajo la curva normal 735
Tabla A.3 Áreas bajo la curva normal 0 z
Área
 z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
 −3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
 −3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
 −3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
 −3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
 −3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
 −2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
 −2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
 −2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
 −2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
 −2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
 −2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
 −2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
 −2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
 −2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
 −2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
 −1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
 −1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
 −1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
 −1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
 −1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
 −1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
 −1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
 −1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
 −1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
 −1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
 −0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
 −0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
 −0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
 −0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
 −0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
 −0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121
 −0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
 −0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
 −0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
 −0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
736 Apéndice A Tablas y demostraciones estadísticas
Tabla A.3 (continuación) Áreas bajo la curva normal
 z .00 .01 .02 .03 .04 .05 .06 .07 .08 .09
 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.57140.5753
 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
 2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
 2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
 2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
 2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
 2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
 2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
 2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
 2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
 2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
 2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
 3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990
 3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
 3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
 3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
 3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
Tabla A.4 Valores críticos de la distribución t 737
Tabla A.4 Valores críticos de la distribución t 0 t
α
α
α
 v 0.40 0.30 0.20 0.15 0.10 0.05 0.025
 1 0.325 0.727 1.376 1.963 3.078 6.314 12.706
 2 0.289 0.617 1.061 1.386 1.886 2.920 4.303
 3 0.277 0.584 0.978 1.250 1.638 2.353 3.182
 4 0.271 0.569 0.941 1.190 1.533 2.132 2.776
 5 0.267 0.559 0.920 1.156 1.476 2.015 2.571
 6 0.265 0.553 0.906 1.134 1.440 1.943 2.447
 7 0.263 0.549 0.896 1.119 1.415 1.895 2.365
 8 0.262 0.546 0.889 1.108 1.397 1.860 2.306
 9 0.261 0.543 0.883 1.100 1.383 1.833 2.262
 10 0.260 0.542 0.879 1.093 1.372 1.812 2.228
 11 0.260 0.540 0.876 1.088 1.363 1.796 2.201
 12 0.259 0.539 0.873 1.083 1.356 1.782 2.179
 13 0.259 0.538 0.870 1.079 1.350 1.771 2.160
 14 0.258 0.537 0.868 1.076 1.345 1.761 2.145
 15 0.258 0.536 0.866 1.074 1.341 1.753 2.131
 16 0.258 0.535 0.865 1.071 1.337 1.746 2.120
 17 0.257 0.534 0.863 1.069 1.333 1.740 2.110
 18 0.257 0.534 0.862 1.067 1.330 1.734 2.101
 19 0.257 0.533 0.861 1.066 1.328 1.729 2.093
 20 0.257 0.533 0.860 1.064 1.325 1.725 2.086
 21 0.257 0.532 0.859 1.063 1.323 1.721 2.080
 22 0.256 0.532 0.858 1.061 1.321 1.717 2.074
 23 0.256 0.532 0.858 1.060 1.319 1.714 2.069
 24 0.256 0.531 0.857 1.059 1.318 1.711 2.064
 25 0.256 0.531 0.856 1.058 1.316 1.708 2.060
 26 0.256 0.531 0.856 1.058 1.315 1.706 2.056
 27 0.256 0.531 0.855 1.057 1.314 1.703 2.052
 28 0.256 0.530 0.855 1.056 1.313 1.701 2.048
 29 0.256 0.530 0.854 1.055 1.311 1.699 2.045
 30 0.256 0.530 0.854 1.055 1.310 1.697 2.042
 40 0.255 0.529 0.851 1.050 1.303 1.684 2.021
 60 0.254 0.527 0.848 1.045 1.296 1.671 2.000
 120 0.254 0.526 0.845 1.041 1.289 1.658 1.980
 ∞ 0.253 0.524 0.842 1.036 1.282 1.645 1.960
738 Apéndice A Tablas y demostraciones estadísticas
Tabla A.4 (continuación) Valores críticos de la distribución t
α
 v 0.02 0.015 0.01 0.0075 0.005 0.0025 0.0005
 1 15.894 21.205 31.821 42.433 63.656 127.321 636.578
 2 4.849 5.643 6.965 8.073 9.925 14.089 31.600
 3 3.482 3.896 4.541 5.047 5.841 7.453 12.924
 4 2.999 3.298 3.747 4.088 4.604 5.598 8.610
 5 2.757 3.003 3.365 3.634 4.032 4.773 6.869
 6 2.612 2.829 3.143 3.372 3.707 4.317 5.959
 7 2.517 2.715 2.998 3.203 3.499 4.029 5.408
 8 2.449 2.634 2.896 3.085 3.355 3.833 5.041
 9 2.398 2.574 2.821 2.998 3.250 3.690 4.781
 10 2.359 2.527 2.764 2.932 3.169 3.581 4.587
 11 2.328 2.491 2.718 2.879 3.106 3.497 4.437
 12 2.303 2.461 2.681 2.836 3.055 3.428 4.318
 13 2.282 2.436 2.650 2.801 3.012 3.372 4.221
 14 2.264 2.415 2.624 2.771 2.977 3.326 4.140
 15 2.249 2.397 2.602 2.746 2.947 3.286 4.073
 16 2.235 2.382 2.583 2.724 2.921 3.252 4.015
 17 2.224 2.368 2.567 2.706 2.898 3.222 3.965
 18 2.214 2.356 2.552 2.689 2.878 3.197 3.922
 19 2.205 2.346 2.539 2.674 2.861 3.174 3.883
 20 2.197 2.336 2.528 2.661 2.845 3.153 3.850
 21 2.189 2.328 2.518 2.649 2.831 3.135 3.819
 22 2.183 2.320 2.508 2.639 2.819 3.119 3.792
 23 2.177 2.313 2.500 2.629 2.807 3.104 3.768
 24 2.172 2.307 2.492 2.620 2.797 3.091 3.745
 25 2.167 2.301 2.485 2.612 2.787 3.078 3.725
 26 2.162 2.296 2.479 2.605 2.779 3.067 3.707
 27 2.158 2.291 2.473 2.598 2.771 3.057 3.689
 28 2.154 2.286 2.467 2.592 2.763 3.047 3.674
 29 2.150 2.282 2.462 2.586 2.756 3.038 3.660
 30 2.147 2.278 2.457 2.581 2.750 3.030 3.646
 40 2.123 2.250 2.423 2.542 2.704 2.971 3.551
 60 2.099 2.223 2.390 2.504 2.660 2.915 3.460
 120 2.076 2.196 2.358 2.468 2.617 2.860 3.373
 ∞ 2.054 2.170 2.326 2.432 2.576 2.807 3.290
Tabla A.5 Valores críticos de la distribución chi cuadrada 739
Tabla A.5 Valores críticos de la distribución chi cuadrada 0 2χ
α
α
α
 v 0.995 0.99 0.98 0.975 0.95 0.90 0.80 0.75 0.70 0.50
 1 0.04393 0.03157 0.03628 0.03982 0.00393 0.0158 0.0642 0.102 0.148 0.455
 2 0.0100 0.0201 0.0404 0.0506 0.103 0.211 0.446 0.575 0.713 1.386
 3 0.0717 0.115 0.185 0.216 0.352 0.584 1.005 1.213 1.424 2.366
 4 0.207 0.297 0.429 0.484 0.711 1.064 1.649 1.923 2.195 3.357
 5 0.412 0.554 0.752 0.831 1.145 1.610 2.343 2.675 3.000 4.351
 6 0.676 0.872 1.134 1.237 1.635 2.204 3.070 3.455 3.828 5.348
 7 0.989 1.239 1.564 1.690 2.167 2.833 3.822 4.255 4.671 6.346
 8 1.344 1.647 2.032 2.180 2.733 3.490 4.594 5.071 5.527 7.344
 9 1.735 2.088 2.532 2.700 3.325 4.168 5.380 5.899 6.393 8.343
 10 2.156 2.558 3.059 3.247 3.940 4.865 6.179 6.737 7.267 9.342
 11 2.603 3.053 3.609 3.816 4.575 5.578 6.989 7.584 8.148 10.341
 12 3.074 3.571 4.178 4.404 5.226 6.304 7.807 8.438 9.034 11.340
 13 3.565 4.107 4.765 5.009 5.892 7.041 8.634 9.299 9.926 12.340
 14 4.075 4.660 5.368 5.629 6.571 7.790 9.467 10.165 10.821 13.339
 15 4.601 5.229 5.985 6.262 7.261 8.547 10.307 11.037 11.721 14.339
 16 5.142 5.812 6.614 6.908 7.962 9.312 11.152 11.912 12.624 15.338
 17 5.697 6.408 7.255 7.564 8.672 10.085 12.002 12.792 13.531 16.338
 18 6.265 7.015 7.906 8.231 9.390 10.865 12.857 13.675 14.440 17.338
 19 6.844 7.633 8.567 8.907 10.117 11.651 13.716 14.562 15.352 18.338
 20 7.434 8.260 9.237 9.591 10.851 12.443 14.578 15.452 16.266 19.337
 21 8.034 8.897 9.915 10.283 11.591 13.240 15.445 16.344 17.182 20.337
 22 8.643 9.542 10.600 10.982 12.338 14.041 16.314 17.240 18.101 21.337
 23 9.260 10.196 11.293 11.689 13.091 14.848 17.187 18.137 19.021 22.337
 24 9.88610.856 11.992 12.401 13.848 15.659 18.062 19.037 19.943 23.337
 25 10.520 11.524 12.697 13.120 14.611 16.473 18.940 19.939 20.867 24.337
 26 11.160 12.198 13.409 13.844 15.379 17.292 19.820 20.843 21.792 25.336
 27 11.808 12.878 14.125 14.573 16.151 18.114 20.703 21.749 22.719 26.336
 28 12.461 13.565 14.847 15.308 16.928 18.939 21.588 22.657 23.647 27.336
 29 13.121 14.256 15.574 16.047 17.708 19.768 22.475 23.567 24.577 28.336
 30 13.787 14.953 16.306 16.791 18.493 20.599 23.364 24.478 25.508 29.336
 40 20.707 22.164 23.838 24.433 26.509 29.051 32.345 33.66 34.872 39.335
 50 27.991 29.707 31.664 32.357 34.764 37.689 41.449 42.942 44.313 49.335
 60 35.534 37.485 39.699 40.482 43.188 46.459 50.641 52.294 53.809 59.335
740 Apéndice A Tablas y demostraciones estadísticas
Tabla A.5 (continuación) Valores críticos de la distribución chi cuadrada
α
 v 0.30 0.25 0.20 0.10 0.05 0.025 0.02 0.01 0.005 0.001
 1 1.074 1.323 1.642 2.706 3.841 5.024 5.412 6.635 7.879 10.827
 2 2.408 2.773 3.219 4.605 5.991 7.378 7.824 9.210 10.597 13.815
 3 3.665 4.108 4.642 6.251 7.815 9.348 9.837 11.345 12.838 16.266
 4 4.878 5.385 5.989 7.779 9.488 11.143 11.668 13.277 14.860 18.466
 5 6.064 6.626 7.289 9.236 11.070 12.832 13.388 15.086 16.750 20.515
 6 7.231 7.841 8.558 10.645 12.592 14.449 15.033 16.812 18.548 22.457
 7 8.383 9.037 9.803 12.017 14.067 16.013 16.622 18.475 20.278 24.321
 8 9.524 10.219 11.030 13.362 15.507 17.535 18.168 20.090 21.955 26.124
 9 10.656 11.389 12.242 14.684 16.919 19.023 19.679 21.666 23.589 27.877
 10 11.781 12.549 13.442 15.987 18.307 20.483 21.161 23.209 25.188 29.588
 11 12.899 13.701 14.631 17.275 19.675 21.920 22.618 24.725 26.757 31.264
 12 14.011 14.845 15.812 18.549 21.026 23.337 24.054 26.217 28.300 32.909
 13 15.119 15.984 16.985 19.812 22.362 24.736 25.471 27.688 29.819 34.527
 14 16.222 17.117 18.151 21.064 23.685 26.119 26.873 29.141 31.319 36.124
 15 17.322 18.245 19.311 22.307 24.996 27.488 28.259 30.578 32.801 37.698
 16 18.418 19.369 20.465 23.542 26.296 28.845 29.633 32.000 34.267 39.252
 17 19.511 20.489 21.615 24.769 27.587 30.191 30.995 33.409 35.718 40.791
 18 20.601 21.605 22.760 25.989 28.869 31.526 32.346 34.805 37.156 42.312
 19 21.689 22.718 23.900 27.204 30.144 32.852 33.687 36.191 38.582 43.819
 20 22.775 23.828 25.038 28.412 31.410 34.170 35.020 37.566 39.997 45.314
 21 23.858 24.935 26.171 29.615 32.671 35.479 36.343 38.932 41.401 46.796
 22 24.939 26.039 27.301 30.813 33.924 36.781 37.659 40.289 42.796 48.268
 23 26.018 27.141 28.429 32.007 35.172 38.076 38.968 41.638 44.181 49.728
 24 27.096 28.241 29.553 33.196 36.415 39.364 40.270 42.980 45.558 51.179
 25 28.172 29.339 30.675 34.382 37.652 40.646 41.566 44.314 46.928 52.619
 26 29.246 30.435 31.795 35.563 38.885 41.923 42.856 45.642 48.290 54.051
 27 30.319 31.528 32.912 36.741 40.113 43.195 44.140 46.963 49.645 55.475
 28 31.391 32.620 34.027 37.916 41.337 44.461 45.419 48.278 50.994 56.892
 29 32.461 33.711 35.139 39.087 42.557 45.722 46.693 49.588 52.335 58.301
 30 33.530 34.800 36.250 40.256 43.773 46.979 47.962 50.892 53.672 59.702
 40 44.165 45.616 47.269 51.805 55.758 59.342 60.436 63.691 66.766 73.403
 50 54.723 56.334 58.164 63.167 67.505 71.420 72.613 76.154 79.490 86.660
 60 65.226 66.981 68.972 74.397 79.082 83.298 84.58 88.379 91.952 99.608
Tabla A.6 Valores críticos de la distribución F 741
Tabla A.6 Valores críticos de la distribución F 0 f
α
α
f
0.05
(v
1
, v
2
)
v
1
 v
2
 1 2 3 4 5 6 7 8 9
 1 161.45 199.50 215.71 224.58 230.16 233.99 236.77 238.88 240.54
 2 18.51 19.00 19.16 19.25 19.30 19.33 19.35 19.37 19.38
 3 10.13 9.55 9.28 9.12 9.01 8.94 8.89 8.85 8.81
 4 7.71 6.94 6.59 6.39 6.26 6.16 6.09 6.04 6.00
 5 6.61 5.79 5.41 5.19 5.05 4.95 4.88 4.82 4.77
 6 5.99 5.14 4.76 4.53 4.39 4.28 4.21 4.15 4.10
 7 5.59 4.74 4.35 4.12 3.97 3.87 3.79 3.73 3.68
 8 5.32 4.46 4.07 3.84 3.69 3.58 3.50 3.44 3.39
 9 5.12 4.26 3.86 3.63 3.48 3.37 3.29 3.23 3.18
 10 4.96 4.10 3.71 3.48 3.33 3.22 3.14 3.07 3.02
 11 4.84 3.98 3.59 3.36 3.20 3.09 3.01 2.95 2.90
 12 4.75 3.89 3.49 3.26 3.11 3.00 2.91 2.85 2.80
 13 4.67 3.81 3.41 3.18 3.03 2.92 2.83 2.77 2.71
 14 4.60 3.74 3.34 3.11 2.96 2.85 2.76 2.70 2.65
 15 4.54 3.68 3.29 3.06 2.90 2.79 2.71 2.64 2.59
 16 4.49 3.63 3.24 3.01 2.85 2.74 2.66 2.59 2.54
 17 4.45 3.59 3.20 2.96 2.81 2.70 2.61 2.55 2.49
 18 4.41 3.55 3.16 2.93 2.77 2.66 2.58 2.51 2.46
 19 4.38 3.52 3.13 2.90 2.74 2.63 2.54 2.48 2.42
 20 4.35 3.49 3.10 2.87 2.71 2.60 2.51 2.45 2.39
 21 4.32 3.47 3.07 2.84 2.68 2.57 2.49 2.42 2.37
 22 4.30 3.44 3.05 2.82 2.66 2.55 2.46 2.40 2.34
 23 4.28 3.42 3.03 2.80 2.64 2.53 2.44 2.37 2.32
 24 4.26 3.40 3.01 2.78 2.62 2.51 2.42 2.36 2.30
 25 4.24 3.39 2.99 2.76 2.60 2.49 2.40 2.34 2.28
 26 4.23 3.37 2.98 2.74 2.59 2.47 2.39 2.32 2.27
 27 4.21 3.35 2.96 2.73 2.57 2.46 2.37 2.31 2.25
 28 4.20 3.34 2.95 2.71 2.56 2.45 2.36 2.29 2.24
 29 4.18 3.33 2.93 2.70 2.55 2.43 2.35 2.28 2.22
 30 4.17 3.32 2.92 2.69 2.53 2.42 2.33 2.27 2.21
 40 4.08 3.23 2.84 2.61 2.45 2.34 2.25 2.18 2.12
 60 4.00 3.15 2.76 2.53 2.37 2.25 2.17 2.10 2.04
 120 3.92 3.07 2.68 2.45 2.29 2.18 2.09 2.02 1.96
 ∞ 3.84 3.00 2.60 2.37 2.21 2.10 2.01 1.94 1.88
Reproducida de la tabla 18 de Biometrika Tables for Statisticians, Vol. I, con autorización de E.S. Pearson 
y Biometrika Trustees.
742 Apéndice A Tablas y demostraciones estadísticas
Tabla A.6 (continuación) Valores críticos de la distribución F
f
0.05
(v
1
, v
2
)
v
1
 v
2
 10 12 15 20 24 30 40 60 120 ∞
 1 241.88 243.91 245.95 248.01 249.05 250.10 251.14 252.20 253.25 254.31
 2 19.40 19.41 19.43 19.45 19.45 19.46 19.47 19.48 19.49 19.50
 3 8.79 8.74 8.70 8.66 8.64 8.62 8.59 8.57 8.55 8.53
 4 5.96 5.91 5.86 5.80 5.77 5.75 5.72 5.69 5.66 5.63
 5 4.74 4.68 4.62 4.56 4.53 4.50 4.46 4.43 4.40 4.36
 6 4.06 4.00 3.94 3.87 3.84 3.81 3.77 3.74 3.70 3.67
 7 3.64 3.57 3.51 3.44 3.41 3.38 3.34 3.30 3.27 3.23
 8 3.35 3.28 3.22 3.15 3.12 3.08 3.04 3.01 2.97 2.93
 9 3.14 3.07 3.01 2.94 2.90 2.86 2.83 2.79 2.75 2.71
 10 2.98 2.91 2.85 2.77 2.74 2.70 2.66 2.62 2.58 2.54
 11 2.85 2.79 2.72 2.65 2.61 2.57 2.53 2.49 2.45 2.40
 12 2.75 2.69 2.62 2.54 2.51 2.47 2.43 2.38 2.34 2.30
 13 2.67 2.60 2.53 2.46 2.42 2.38 2.34 2.30 2.25 2.21
 14 2.60 2.53 2.46 2.39 2.35 2.31 2.27 2.22 2.18 2.13
 15 2.54 2.48 2.40 2.33 2.29 2.25 2.20 2.16 2.11 2.07
 16 2.49 2.42 2.35 2.28 2.24 2.19 2.15 2.11 2.06 2.01
 17 2.45 2.38 2.31 2.23 2.19 2.15 2.10 2.06 2.01 1.96
 18 2.41 2.34 2.27 2.19 2.15 2.11 2.06 2.02 1.97 1.92
 19 2.38 2.31 2.23 2.16 2.11 2.07 2.03 1.98 1.93 1.88
 20 2.35 2.28 2.20 2.12 2.08 2.04 1.99 1.95 1.90 1.84
 21 2.32 2.25 2.18 2.10 2.05 2.01 1.96 1.92 1.87 1.81
 22 2.30 2.23 2.15 2.07 2.03 1.98 1.94 1.89 1.84 1.78
 23 2.27 2.20 2.13 2.05 2.01 1.96 1.91 1.86 1.81 1.76
 24 2.25 2.18 2.11 2.03 1.98 1.94 1.89 1.84 1.79 1.73
 25 2.24 2.16 2.09 2.01 1.96 1.92 1.87 1.82 1.77 1.71
 26 2.22 2.15 2.07 1.99 1.95 1.90 1.85 1.80 1.75 1.69
 27 2.20 2.13 2.06 1.97 1.93 1.88 1.84 1.79 1.73 1.67
 28 2.19 2.12 2.04 1.96 1.91 1.87 1.82 1.77 1.71 1.65
 29 2.18 2.10 2.03 1.94 1.90 1.85 1.81 1.75 1.70 1.64
 30 2.16 2.09 2.01 1.93 1.89 1.84 1.79 1.74 1.68 1.62
 40 2.08 2.00 1.92 1.84 1.79 1.74 1.69 1.64 1.58 1.51
 60 1.99 1.92 1.84 1.75 1.70 1.65 1.59 1.53 1.47 1.39
 120 1.91 1.83 1.75 1.66 1.61 1.55 1.50 1.43 1.35 1.25
 ∞ 1.83 1.75 1.67 1.57 1.52 1.46 1.39 1.32 1.22 1.00
Tabla A.6 Valores críticos de la distribución F 743
Tabla A.6 (continuación) Valores críticos de la distribución F
f
0.01
(v
1
, v
2
)
v
1
 v
2
 1 2 3 4 5 6 7 8 9
 1 4052.18 4999.50 5403.35 5624.58 5763.65 5858.99 5928.36 5981.07 6022.47
 2 98.50 99.00 99.17 99.25 99.30 99.33 99.36 99.37 99.39
 3 34.12 30.82 29.46 28.71 28.24 27.91 27.67 27.49 27.35
 4 21.20 18.00 16.69 15.98 15.52 15.21 14.98 14.80 14.66
 5 16.26 13.27 12.06 11.39 10.97 10.67 10.46 10.29 10.16
 6 13.75 10.92 9.78 9.15 8.75 8.47 8.26 8.10 7.98
 7 12.25 9.55 8.45 7.85 7.46 7.19 6.99 6.84 6.72
 8 11.26 8.65 7.59 7.01 6.63 6.37 6.18 6.03 5.91
 9 10.56 8.02 6.996.42 6.06 5.80 5.61 5.47 5.35
 10 10.04 7.56 6.55 5.99 5.64 5.39 5.20 5.06 4.94
 11 9.65 7.21 6.22 5.67 5.32 5.07 4.89 4.74 4.63
 12 9.33 6.93 5.95 5.41 5.06 4.82 4.64 4.50 4.39
 13 9.07 6.70 5.74 5.21 4.86 4.62 4.44 4.30 4.19
 14 8.86 6.51 5.56 5.04 4.69 4.46 4.28 4.14 4.03
 15 8.68 6.36 5.42 4.89 4.56 4.32 4.14 4.00 3.89
 16 8.53 6.23 5.29 4.77 4.44 4.20 4.03 3.89 3.78
 17 8.40 6.11 5.18 4.67 4.34 4.10 3.93 3.79 3.68
 18 8.29 6.01 5.09 4.58 4.25 4.01 3.84 3.71 3.60
 19 8.18 5.93 5.01 4.50 4.17 3.94 3.77 3.63 3.52
 20 8.10 5.85 4.94 4.43 4.10 3.87 3.70 3.56 3.46
 21 8.02 5.78 4.87 4.37 4.04 3.81 3.64 3.51 3.40
 22 7.95 5.72 4.82 4.31 3.99 3.76 3.59 3.45 3.35
 23 7.88 5.66 4.76 4.26 3.94 3.71 3.54 3.41 3.30
 24 7.82 5.61 4.72 4.22 3.90 3.67 3.50 3.36 3.26
 25 7.77 5.57 4.68 4.18 3.85 3.63 3.46 3.32 3.22
 26 7.72 5.53 4.64 4.14 3.82 3.59 3.42 3.29 3.18
 27 7.68 5.49 4.60 4.11 3.78 3.56 3.39 3.26 3.15
 28 7.64 5.45 4.57 4.07 3.75 3.53 3.36 3.23 3.12
 29 7.60 5.42 4.54 4.04 3.73 3.50 3.33 3.20 3.09
 30 7.56 5.39 4.51 4.02 3.70 3.47 3.30 3.17 3.07
 40 7.31 5.18 4.31 3.83 3.51 3.29 3.12 2.99 2.89
 60 7.08 4.98 4.13 3.65 3.34 3.12 2.95 2.82 2.72
 120 6.85 4.79 3.95 3.48 3.17 2.96 2.79 2.66 2.56
 ∞ 6.63 4.61 3.78 3.32 3.02 2.80 2.64 2.51 2.41
744 Apéndice A Tablas y demostraciones estadísticas
Tabla A.6 (continuación) Valores críticos de la distribución F
f
0.01
(v
1
, v
2
)
v
1
 v
2
 10 12 15 20 24 30 40 60 120 ∞
 1 6055.85 6106.32 6157.28 6208.73 6234.63 6260.65 6286.78 6313.03 6339.39 6365.86
 2 99.40 99.42 99.43 99.45 99.46 99.47 99.47 99.48 99.49 99.50
 3 27.23 27.05 26.87 26.69 26.60 26.50 26.41 26.32 26.22 26.13
 4 14.55 14.37 14.20 14.02 13.93 13.84 13.75 13.65 13.56 13.46
 5 10.05 9.89 9.72 9.55 9.47 9.38 9.29 9.20 9.11 9.02
 6 7.87 7.72 7.56 7.40 7.31 7.23 7.14 7.06 6.97 6.88
 7 6.62 6.47 6.31 6.16 6.07 5.99 5.91 5.82 5.74 5.65
 8 5.81 5.67 5.52 5.36 5.28 5.20 5.12 5.03 4.95 4.86
 9 5.26 5.11 4.96 4.81 4.73 4.65 4.57 4.48 4.40 4.31
 10 4.85 4.71 4.56 4.41 4.33 4.25 4.17 4.08 4.00 3.91
 11 4.54 4.40 4.25 4.10 4.02 3.94 3.86 3.78 3.69 3.60
 12 4.30 4.16 4.01 3.86 3.78 3.70 3.62 3.54 3.45 3.36
 13 4.10 3.96 3.82 3.66 3.59 3.51 3.43 3.34 3.25 3.17
 14 3.94 3.80 3.66 3.51 3.43 3.35 3.27 3.18 3.09 3.00
 15 3.80 3.67 3.52 3.37 3.29 3.21 3.13 3.05 2.96 2.87
 16 3.69 3.55 3.41 3.26 3.18 3.10 3.02 2.93 2.84 2.75
 17 3.59 3.46 3.31 3.16 3.08 3.00 2.92 2.83 2.75 2.65
 18 3.51 3.37 3.23 3.08 3.00 2.92 2.84 2.75 2.66 2.57
 19 3.43 3.30 3.15 3.00 2.92 2.84 2.76 2.67 2.58 2.49
 20 3.37 3.23 3.09 2.94 2.86 2.78 2.69 2.61 2.52 2.42
 21 3.31 3.17 3.03 2.88 2.80 2.72 2.64 2.55 2.46 2.36
 22 3.26 3.12 2.98 2.83 2.75 2.67 2.58 2.50 2.40 2.31
 23 3.21 3.07 2.93 2.78 2.70 2.62 2.54 2.45 2.35 2.26
 24 3.17 3.03 2.89 2.74 2.66 2.58 2.49 2.40 2.31 2.21
 25 3.13 2.99 2.85 2.70 2.62 2.54 2.45 2.36 2.27 2.17
 26 3.09 2.96 2.81 2.66 2.58 2.50 2.42 2.33 2.23 2.13
 27 3.06 2.93 2.78 2.63 2.55 2.47 2.38 2.29 2.20 2.10
 28 3.03 2.90 2.75 2.60 2.52 2.44 2.35 2.26 2.17 2.06
 29 3.00 2.87 2.73 2.57 2.49 2.41 2.33 2.23 2.14 2.03
 30 2.98 2.84 2.70 2.55 2.47 2.39 2.30 2.21 2.11 2.01
 40 2.80 2.66 2.52 2.37 2.29 2.20 2.11 2.02 1.92 1.80
 60 2.63 2.50 2.35 2.20 2.12 2.03 1.94 1.84 1.73 1.60
 120 2.47 2.34 2.19 2.03 1.95 1.86 1.76 1.66 1.53 1.38
 ∞ 2.32 2.18 2.04 1.88 1.79 1.70 1.59 1.47 1.32 1.00